Depends on the genre you’re looking at. (Just forgive my formatting!) :o
Planet Puzzle League - This is your classic Tetris Attack or Panel de Pon–a match-three game exponentially deeper than Bejeweled–but with touch controls, time trials, and online play.
Tetris DS - Same as above, but with Tetris. Features awesome Nintendo retro themes during gameplay, as well as some new modes and one that takes advantage of the touch screen. This one’s rather rare, tho.
Meteos - An early touch-based puzzler where the blocks you match serve as rockets to the top of the screen, where they disappear. The physics for each level are different, so you may have to make smaller matches in order for them to beat the level’s gravity.
**Peggle Dual Shot **- It’s fucking PEGGLE! And Peggle Nights, so it’s two games in one.
**Gunpey DS **- Found this in a bargain bin and ended up loving it. It’s a puzzle game where you shift around blocks with angled lines on them, and when the lines match up in some way that makes an unbroken connection from the left to the right of the screen, you get a bonus and those blocks disappear. It also moves pretty well to the music.
**Henry Hatsworth **- This latest from EA needs to be played; it’s a shame most people won’t give it a chance because it doesn’t already have a big-brand name attached to it.
New Super Mario Bros - It’s sold millions for a reason. This game brings back all the classic moves, enemies, tropes, and level types from classic 2D Mario games into a new 8 World adventure. They need to hurry up and make a sequel to it.
**Castlevania: Dawn of Sorrow **- A comparatively easy Castlevania (of the SoTN type, focusing on exploration) with a twist: every time you kill an enemy, you have a chance of getting its “soul,” which you can equip and use as a unique power. These can range from simply throwing a projectile, summoning a huge sword to help you, to giving you the ability to fly. Some souls are super-rare, so it’s almost got that Pokemon-like collecting madness to it.
**Legend of Zelda: Phantom Hourglass **- An entirely touch-based Zelda game, which controls surprisingly well for what it is. You slash near Link to attack, certain puzzles become much easier with the game’s built-in map drawing system, and things like bombs and boomerangs are suddenly much more accurate. As always, much to explore and discover.
The World Ends With You - Even with the crazy anime look to it, this is one of the more maturely written RPGs on the system. It’s set in a heavily stylized version of the Shibuya district of Tokyo, which fits with the electro-sounding J-rock soundtrack. The combat is a real test, as it has you controlling one character on the top screen with the buttons and using gesture-based attacks on the bottom screen. You can play an easy mode, but the game features different unlocks and skills on higher settings and lets you adjust the difficulty pretty much at will. A lot of experimental stuff in this game, but it ends up working out to make something unique in both the genre and the handheld market.
Chrono Trigger - Hard to say more than has already been said about this one. One of the first active-battle RPGs, where enemies will attack at regular intervals even while you’re thinking about what attack to select. Too many great characters, moments, and sidequests to mention. Only downside is they haven’t changed much of it.
Final Fantasy IV - This one they changed, from the translation to the graphics to the minute rewards for exploring every inch of a dungeon. It was born long ago as one of the consummate Final Fantasy games and one of the very last strictly turn-based ones. Lots of grinding is necessary at some points. It’s a 2D game fully realized in 3D, so even without the great story, it’s a marvel just for that.
Izuna: Legend of the Unemployed Ninja / Shiren the Wanderer / FF: Chocobo’s Dungeon - These are worth a try if you’ve ever been sucked into a roguelike such as Nethack or ADOM. They use the same elements such as randomly generated dungeons, tile-based movement with turn-based attacks, and rather sharp penalties for death. Generally more lenient than PC roguelikes, but still a great way to sink lots of time, if you have the patience.
Phoenix Wright: Ace Attorney - Unrealistic courtroom drama. Lots of finger-pointing and zealous arguments. You play a defense attorney who, in the process of learning about his client’s case, collects evidence and pieces it together to find the real killer. There are always a few twists for each case, so it often comes down to the last day in court and a key witness revealing how the story plays out. The writing and presentation are great, tho the linear one-way logic can be frustrating, especially when there’s a lot of evidence pointing toward a particular conclusion, but you’re only allowed to present one piece at a time.
Hotel Dusk - Blend touch-based puzzles with a film-noir feel about a mysterious hotel room.
**Brain Age **- Another easy million-seller. It ostensibly uses a polygonal Japanese doctor’s tendency to lecture you about breakfast to make your brain feel younger, but it turns out to be more about unlocking new brain exercises and rewarding you for sticking with it and playing every day. Also includes a pretty good sudoku.
The DS has some great games out, if you know what to look for, and is definitely one of the easiest systems to get into. Hope these help!